You have never told us completely and exactly what the problem is. Nor have you given us any clue about your level of mathematical knowledge. Consequently, it is very difficult to give you an answer that is both relevant and understandable by you.

You graph functions, not equations. Now it is very remotely possible that the mess you are given defines IMPLICITLY a single function in one independent and one dependent variable, but in all probability it defines implicitly a number of different functions with one independent variable. Determing what function or functions can be restated as explicit functions in two variables may be theoretically conceivable, but it looks dauntingly ugly even if possible.

However, you can also think about the expression on the left as being an explicit function with two independent variables and one dependent variable. I greatly doubt that there is any practical way to graph that three dimensional object in a comprehensible way.

Finally, you can think about this as an equation and forget all about graphing. It is easy to find solutions to the equation. What may be difficult is finding all the solutions.

$\{f(x,\ y)\}^2 + \{g(y)\}^2 = 0 \implies f\{(x,\ y)\}^2 = -\ \{g(y)\}^2.$

Now if x and y are supposed to be real numbers f(x, y) and g(y) are real numbers, which means that g(y) = 0. That lets you solve for y.

Do you understand what I am saying?